Journal of Applied Finance & Banking, vol.1, no.1, 2011, 53-81 ISSN: 1792-6580 (print version), 1792-6599 (online) International Scientific Press, 2011 Market timing and statistical arbitrage: Which market timing opportunities arise from equity price busts coinciding with recessions? The Swedish stock market in the financial crises 2008 Klaus Grobys 1 Abstract Even though a random walk process is from a statistical point of view not predictable, some movements can be correlated with specific events concerning other variables. Then, predictable patterns may arise being dependent on this joint event. There is evidence given that equity price busts being associated with recessions continue until the economy switches from the state of recession to an economic pick-up. The following contribution takes into account the Swedish stock index OMX 30 and 25 preselected stocks. The out-of-sample period runs from September 12, 2008 – March 12, 2009, whereas on September 11, 2008 the official press release was issued that European economies face a recession. This study suggests a market timing opportunity resulting in a maximum statistical arbitrage opportunity corresponding to a profit of 19% p.a. with an empirical probability of 50.14%. The optimal defensive strategies, however, exhibit excess 1 E-mail: [email protected]Article Info: Revised: March 5, 2011. Published online : May 31, 2011
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The perception that stock prices already reflect all available information being
often referred to as the efficient market hypothesis is widely discussed in the
academic literature and rests upon studies by Kendall (1953). As soon as there is
any news available indicating that a stock is underpriced, Bodie, Kane and Marcus
(2008) highlight that rational investors would buy this stock immediately and,
hence, bid up its price to a level where only ordinary returns can be expected to
gain. Consequently, the efficient market hypothesis suggests that stock prices
follow random walk processes involving that prices changes are random and thus
unpredictable. Studies by Chan, Gup and Pan (1997) who examine eighteen
national stock markets by using unit root tests figure out that the world equity
markets are weak-form efficient and, hence, support the efficient market
hypothesis. Unit-root tests as applied by Chan, Gup and Pan (1997) though take
only information into account which is involved in the univariate data generating
processes.
Against this, Claessens, Kose and Terrones (2009) provide a comprehensive
empirical characterization of linkages between key macroeconomic and financial
variables around business and financial cycles. Their studies involve 21 OECD
countries and cover over 47 years from 1960-2007. Thereby, they take into
account 122 recessions, 113 credit contractions and 245 episodes of equity prices
declines, whereas 61 of these equity price declines are referred to as price busts.
Equity price busts are in accordance to the definition of Claessens, Kose and
K. Grobys 55
Terrones (2009) peak-to-trough declines in equity prices which fall within the top
quartile of all price declines. Their findings show that equity price busts overlap
about one-third of the recession episode. Furthermore, given the event that the
economy faces a recession, in 60% of all cases equity price declines occur at the
same time.
Moreover, Claessens, Kose and Terrones (2009) conclude that recessions tend
to coincide with contractions in domestic credit and declines in asset prices and in
most advanced countries. Thereby, the typical duration of en equity price bust is
twice that of a recession, but ends at the same time when the associated recession
is ending. But what implications do equity price busts offer concerning asset
allocations if the bust coincides with a recession? If market participators expect
stock prices to fall in further periods, they will rebalance their stock portfolios
such that the expected loss will be minimized. As an alternative, investors could
construct arbitrage portfolios while going short on the index and long on an equity
portfolio exhibiting defensive properties and therewith outperforms if the stock
markets declines in future periods.
In the following contribution the equity price bust being associated with the
financial crises in 2008 is analyzed with respect to market-timing opportunities.
Furthermore, the optimization problem being associated with an advantageous
asset allocation conditional on the state of the economy is examined. Thereby, the
Swedish leading stock index OMX 30 is taken into account. 50 different portfolios
are estimated which track artificial indices corresponding to defensive investment
strategies. The optimization procedure accounts for 20 stocks corresponding to the
companies exhibiting the highest market capitalization being line with Alexander
and Dimitriu (2005). While holding the optimal weights constant within a
six-month period out-of-sample (i.e. September 12, 2008-March 12, 2009),
evidence is given for statistical arbitrage opportunities. The estimated optimal
asset allocations as suggested here dominate the index in both, the Reward-to-Risk
ratio and the Reward-to-Risk-Difference ratio. During the six months period being
56 Market timing and statistical arbitrage
examined both events occur at the same time, the economy faces a recession and
the financial market faces an equity price bust. Consequently, a rational investor
who expects the equity prices to fall in future periods will select a defensive asset
allocation in order to minimize expected losses as soon as the recession is
ascertained. Optimal defensive strategies, as suggested here, exhibit, given the
considered Swedish stock market conditions, returns of -19.19% p.a. and -6.13%
p.a. involving a volatility of 53.50%, respectively, 61.50% p.a. The OMX 30
though had a return of -34.16% p.a. and exhibited a volatility being equal to
53.51% under the same period of consideration (i.e. September 12, 2008-March
12, 2009).
2 Background
Even though the efficient market hypothesis holds when testing stock markets
price movements of most advanced countries whether the event "equity price
bust" occurs, given the event that the economy faces a recession, predictable
patterns may evolve. Claessens, Kose and Terrones (2009) who consider a large
data set of recessions, equity price declines and credit contractions within OECD
countries, argue that a typical episode of an equity price decline, respectively, an
equity price bust tends to result in a 24% and 51% fall in equity prices. Thereby,
the duration’s mean is 6.64 and, respectively 11.79 quarters where the latter figure
is statistically significant even on a 1% significance level. Furthermore, recessions
that coincide with equity price busts last for 3.79 quarters on average, whereas
recessions that do not coincide with equity price busts last 3.49 quarters on
average.
Claessens, Kose and Terrones (2009) conclude that equity price declines
overlap with about one in three recessions. If a recession coincides with an equity
price bust, the recession can start as late as four to five quarters after the asset bust
has started. However, the equity price bust typically ends with the end of its
K. Grobys 57
corresponding recession but can continue for two to nine quarters after the
recession has ended. Against it, the minimum duration of a recession is in
accordance to Claessens, Kose and Terrones (2009) two quarters, whereas a
typical recession lasts about four quarters. The latter fact clearly exhibits market
timing potential: A rational investor who expects stock prices to fall during the
next two quarters will choose a defensive asset allocation in order to minimize the
expected loss in future periods. Furthermore, investors could exploit from equity
price declines by shorting the index and taking a position in stocks which response
defensively.
Once the recession is ascertained, a rational market participant will rebalance
the equity portfolio in order to anticipate a further fall in equity prices. Even
though Aroa and Buza (2003) mention that recessions are not periodic and that
they differ in duration, intensity and occurrence, there are still similarities in the
sequence of events and circumstances that typically occur over the course of a
business cycle. The same is the case with respect to stock market crashes. Each
stock market crash is preceded by a bubble formation as argued by Aroa and Buza
(2003) where bubbles, respectively, bull markets are usually associated with a
period of prosperity, when the future seems bright and investors have easy access
to money. Against this, excessive pessimism follows this exuberance and creates
as a consequence the stock market crash, respectively, the bear market. In
accordance to Aroa and Buza (2003), the same mass psychology evoking the
expectation that every dot-com company will be profitable and, hence, created the
boom in the stock market during 1995-1999, was accountable for the crash in the
NASDAQ in January-March 2000. The U.S economy began to slow down during
the second half of the year 2000, and the rest of the world followed, resulting in a
worldwide recession. If the market stands in a bear market, crisis events which can
be generalized as bad news exacerbate the stock market’s downturn movements as
mentioned by Aroa and Buza (2003). Of course, the downturn will not end as long
as the majority of news which arrive the market will be evaluated as good news
58 Market timing and statistical arbitrage
from the market participitiants' point of view. Hence, press releases such as issued
on September 11, 2008 from the German Insitute for Worldeconomics (IfW Kiel)
declaring that European countries face a recession will consequently be associated
with an expectation that stock prices will continue to decline even in future
periods.2
For instance, the Swedish leading stock index OMX 30 lost already 36.91%
compared to its peak on July 16, 2007 on the day where the official press release
was issued (i.e. September 11, 2008). Considering a period of two quarters
thereafter it could be observed that the OMX 30 fell by additional 17.08% (i.e.
from September 12, 2008 – March 12, 2009). Hence, the equity price bust began
more about 14 months before the recession was ascertained and continued
afterwards. The same patterns could be observed in 2001. On November 26, 2001,
the National Bureau of Economic Research issued a press release declaring the
recession began in March 2001.3
Market observers recognized similar patterns: From November 26, 2001 until
May 26, 2002 the Swedish leading stock index OMX 30 fell by 18.85%. However,
the recession was also anticipated by an equity price bust where the OMX 30 lost
already 44.27% from its peak on March 7, 2000 until the day where the recession
was declared by the National Bureau of Economic Research (i.e. on November 26,
2001). The same patterns could be investigated concerning other European stock
markets. For instance, the German's leading stock index DAX fell in the period
November 26, 2001 until May 26, 2002 by 36.89%, whereas the stock index
additionally fell by 4.20% from between November 26, 2001 and May 26, 2002.
Considering the financial crises in 2008 and the associated equity price bust which
again anticipated the recession, the DAX lost 23.77% from July 16, 2007 until
September 11, 2008 and only additional 1.54% from September 12, 2008 – March
12, 2009.
2 See http://www.ifw-kiel.de/media/press-releases/2008/pr11-09-08b.
K. Grobys 59
However, not all stocks participate in booms, respectively, bull markets. In
accordance to Aroa and Buza (2003) railroad stocks were excluded from the boom
of 1928-1929, whereas overinvesting in utilities caused this speculative bubble
formation. The bubble formation during 1995-1999 showed an overpricing of the
telecommunication and internet sector as studied by Jensen (2005) and Harmantzis
(2004), whereas a similar mass psychology caused the overpricing concerning the
financial sector during 2004-2008 as described by Baker (2008) and Soros (2008).
Poterba and Summers (1989) and Cecchetti, Lam and Mark (1990) point out that
the sector will adjust the stronger the more excessive the speculative bubble has
been.
Therefore, a rational investor who expects the market to decline in further
periods will allocate the assets to a portfolio exhibiting defensive properties during
the equity price decline. As a consequence of the equity price bust which started
March 2000, Aroa and Buza (2003) mention that investors had moved the money
into energy and health care company stocks during 2000 and 2001 since these
sectors were expected to response defensively in bear markets. But do defensive
asset allocation strategies being built on historical stochastic movements of
artificial indices exhibit robustness within the out-of-sample period? This
contribution throws light on the following issues: First, 50 different asset
allocations will be estimated which track constructed artificial indices assuming to
exhibit defensive properties if the investors expect the market to decline in further
periods. Thereby, the Swedish stock index OMX 30 will be employed in order to
construct artificial indices and a set of 20 preselected stocks will be used in order
to estimate optimal asset allocation weights. Second, based on these portfolios
tracking defensive artificial indices it will be determined which would be the
optimal asset allocation, given the out-of-sample risk-return estimates. Thereby,
two different optimization calculi will be taken into account. The third issue is that
3 See http://www.nber.org/cycles/november2001/ (accessed on December 02, 2010 21:25).
60 Market timing and statistical arbitrage
it will be discussed how this market-timing approach can be applied for both,
funds management and hedge funds management.
3 Econometric Methodology
In order to estimate weight allocations exhibiting defensive stock portfolios, three
years of historical daily data is taken into account. Following Alexander and
Dimitriu (2005), the cointegration approach is employed where three years of
daily data is necessary to estimate robust cointegration optimal allocation weights.
The day where the press release is issued will in the following be denoted as rect ,
whereas the day where the stock index exhibits the highest notation during the
latest bubble formation will in the following be denoted as maxt . In line with
Grobys (2010) a linear trend is added to the historical index returns that switch the
direction on the day where the price bust begins. Since the exact day is unknown,
it will be assumed that the price bust takes place on day maxt since on the latter
day the stock market notation shows the maximum difference between maxt and rect within the last three years. Arora and Buza (2003) report that not all stocks
participate in bubble formations. Hence, estimating portfolios that do not follow
the market’s exaggeration are expected to decline less than the market during the
crash and can consequently be employed to estimate portfolios involving
defensive asset allocations. In line with Grobys (2010) the linear trend is first
subtracted to the market returns and switches at time point maxt the direction.
Subtracting a linear trend term until maxt and adding the term from observation maxt onwards results in an artificial index being below the benchmark until maxt
and exhibiting higher returns from maxt onwards as the bubble disperses. Then,
the integrated time series corresponding to the artificial indices are given by
1 1 t tOMX
t ti ip c R iδ δ
= == + − ⋅∑ ∑ for max1,...,t t= (1)
max max max t tOMXt tt i t i t
p p R iδ δδ
= == + + ⋅∑ ∑ for max 1,..., rect t t= + , (2)
K. Grobys 61
where δ denotes the factor that is subtracted, respectively, added to the index in
daily terms and OMXtR denotes ordinary index returns at time t . Hence, for each
1 ,..., Mδ δ δ= different integrated artificial indices tpδ can be generated. Figure
(1) shows the index and the artificial index for the factor 0.10δ = (i.e. 25% in
annual terms) for the in-sample period. The integrated time series of stocks being
employed to track the artificial indices are in line with Grobys (2010) calculated
such that
1
rectkt kti
p c R=
= +∑ , (3)
where ktR denotes the return of stock k at time t and c is a constant term.
In order to estimate cointegration optimal weight allocations, the
maximum-likelihood optimization procedure is employed being in line with
Grobys (2010) and given by
( ) ( ) ( )2
22
1log , , log 2 log2 2 2
t
t T
T TL t δεθ δ π σσ∈
⎛ ⎞⎜ ⎟= − ⋅ − −⎜ ⎟⎝ ⎠
∑ (4)
where 1
Kt t k ktk
p a pδ δ δ δε=
= − ⋅∑ . In accordance to van Montefort, Visser and Fijn
van Draat (2008) it is usual to impose weight restrictions. In the following it is
sufficient though to restrict the weights to sum up to one and to be positive being
given by
1Kki k
aδ==∑ (5)
0kaδ > for 1,...,k K= . (6)
62 Market timing and statistical arbitrage
Figure 1: The OMX 30 and the an artificial index within the in-sample period
The weights being estimated at day rect are hold constant two quarters ahead as
the market decline is in accordance to Claessens, Kose and Terrones (2009)
expected to end with the recession while the minimum recession takes per
definition two quarters. Each optimal weight allocation is stored in a vector and
employed to estimate M different out-of-sample portfolio processes depending
on 1 ,..., Mδ δ δ= . The optimal defensive strategies can be determined by
optimizing the two following optimization problems being different from each
other: First, the Reward-to-Risk ratio can be maximized, given by
( )
max1/ 1/
rec rec
rec rec
T T OMXt tt t t t
T Tt tt t t t
R R
N R N R
δ
δδ δ
= =
= =
⎧ ⎫−⎪ ⎪⎨ ⎬⎪ ⎪− ⋅⎩ ⎭
∑ ∑∑ ∑
, (7)
where 1 1ˆ ˆ...t t K KtR a R a Rδ δ δ δ= ⋅ + + ⋅ denotes the estimated returns of the portfolio
δ that tracks the artificial index tpδ , OMXtR denotes the ordinary index returns
and the out-of-sample window runs from 1,...,rect t T= + while recN T t= −
K. Grobys 63
denotes the trading days within the out-of-sample period. In equation (7) it is
calculated how much does an additional return above the benchmark cost in terms
of volatility and rests upon the Reward-to-Risk ratio being introduced by Sharpe
(1964). The maximum value is optimal in the sense that it depicts the asset
allocation that generates the highest return for each unit portfolio volatility with
respect to the out-of-sample time window. However, the optimal asset allocation
can be another one if the excess returns are related to the increase of volatility: In
this case a rational investor would prefer to invest in portfolio ltp instead of
mtp if the increase of excess returns exceeds the increase in portfolio volatility.
Then, the optimization problem is in contrast to equation (7) given by
( )max
1/ 1/ 1/
rec rec
rec rec rec rec
T T OMXt tt t t t
T T N TOMX OMXt t t tt t t t t t t t
R R
N R N R R N R
δ
δδ δ
= =
= = = =
⎧ ⎫−⎪ ⎪⎪ ⎪
⎨ ⎬⎛ ⎞⎪ ⎪− ⋅ − − ⋅⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭
∑ ∑∑ ∑ ∑ ∑
(8)
As the volatility of the optimal portfolio can be lower compared to the stock
market’s volatility, the absolute amount has to be maximized. The constructed
portfolios are tested whether the maximum likelihood estimation provides weight
allocations that exhibit a cointegration relationship with the artificial indices being
tracked concerning the in-sample period. In line with Alexander and Dimitriu
(2005) the ADF-test will be employed, given by
11
ˆ ˆ ˆL
t t l t l tl
uδ δ δ δ δ δε γ ε α ε− −=
Δ = + Δ +∑ (11)
Thereby, the null hypothesis tested is of no cointegration, i.e. 0δγ = ., against the
alternative of 0δγ < . 4 Whether the null hypothesis of no cointegration is
rejected, the cointegration-optimal tracking portfoliod based on the maximum
likelihood procedure of equation (4) is expected to have similar stochastic patterns
4The critical values for the t-statistic of y are obtained using the response surfaces provided by MacKinnon (1991).
64 Market timing and statistical arbitrage
as the artificial indices concerning the in-sample period. The error vector ˆ tδε
comes from an auxiliary regression from the integrated portfolio time series on the
integrated artificial market index times series such that ˆˆ t kt tp pδ δ δ δε β= − ⋅ where,
( ) 1ˆ ' 't t t ktp p p pδ δ δ δ δβ−
= ⋅ ⋅ (see equations (1) and (2)). If ˆ tδε is stationary, the
estimated portfolios are said to be cointegration optimal. Since the artificial
indices are via construction cointegrated with the benchmark, the portfolios can be
considered as being cointegrated with the ordinary benchmark, too.
Furthermore, the out-of-sample portfolios are priced first of all by running
OLS-regressions as following:
OMXt t tR R uδ δ δ δα β= + + , (11)
where tu δ is assumed to be a white noise process and
( )1 1100 /t t t tR p p pδ δ δ δ− −= − ⋅ . Equation (11) is often referred to as ordinary index
model (see Bodie, Kane and Marcus 2008) and is usually employed to determine
whether the portfolio beta (i.e. δβ ) is above or below the market beta being equal
to one. Thereby, a beta being larger than one indicates an offensive asset
allocation while a beta below one usually indicates a defensive one. Furthermore,
if the portfolio alpha (i.e. δα ) is statistically significant higher than zero, the
portfolio is said to generate abnormal returns and, hence, involves statistical
arbitrage opportunities. However, Grobys (2010) mentions that the results of
regressions such as formalized by equation (11) can be misleading as the statistical
arbitrage is cached in the trend-stationary stochastic process being integrated in
the portfolio processes. Moreover, regressions that take into account only the
detrended series, such as the portfolio returns, do not account for this issue as
mentioned by Alexander (1999). In line with Bondarenko (2003) a statistical
arbitrage opportunity arises when the expected payoff of a zero-cost trading
strategy is positive and negative returns occur only stochastically. Therefore, it
will be analyzed how much the empirical probability is that an estimated portfolio
K. Grobys 65
exhibits returns being above the benchmark, that is
( ) ( ) ( )( )1 1,...,OMX rect t t tP E R E R t t Tδ δ≥ + = + .
Finally, a regression is performed in order to figure out how well the
enhancement factors predict the out-of-sample excess returns, respectively,
out-of-sample performance:
( )1/ rec rec
T T OMXt tt t t t
N R R c uδ δδ= =
− = + +∑ ∑ (12)
where uδ is assumed to be a white noise error term with ( )20, uuδ σ∼ , c is a
constant term and ( )rec rec
T T OMXt tt t t t
R Rδ= =−∑ ∑ denotes the excess return of
portfolio corresponding to the enhancement factors 1 ,..., Mδ δ δ= within the
out-of-sample period. If the data suggest a breakpoint, equation (12) is augmented
by a dummy variable accounting for a break in the parameters given by
( ) 1 2 1 21/ rec rec
T T OMXt tt t t t
N R R c c d dδ δδ δ υ= =
− = + + + +∑ ∑ (13)
where δυ is assumed to be a white noise error term 0d = before the break and
1d = otherwise.
4 Results
In this work, the OMX 30 is employed which is the leading stock index in Sweden
and accounts for stocks of the largest 30 companies in accordance to their market
capitalization. The data concerning the in-sample and out-of-sample periods can
be downloaded for free on the index provider’s website
www.nasdaqomxnordic.com. In order to track the constructed artificial indices, 25
stocks exhibiting the highest market capitalization (see the appendix) on
September 2008 are preselected in order to estimate the maximum likelihood
functions. This stock selection approach is in line with Alexander and Dimitriu
(2005) who also select stock in accordance to their market capitalizations. The
66 Market timing and statistical arbitrage
German Research Insitute for Worldeconomis (IfW, Kiel) issued on September 11,
2008 an official press release where it was reported that the Euro area faces a
recession5. At the same time the Swedish leading stock index OMX 30 lost
already 36.91% compared to its peak on July 16, 2007 (i.e. maxtOMX =1.311,87).
Since rect is in this study September 11, 2008, 750 days before the latter date
have to be taken into account in order to estimate the maximum-likelihood
function corresponding to high frequented daily data from September 21, 2005 –
September, 11, 2008. The asset allocation takes place on September 12, 2008 and
the allocation weights are held constant from September 12, 2008 until March 12,
2009 corresponding to 124N = trading days out-of-sample. Claessens, Kose and
Terrones (2009) denote such price declines such as the OMX 30 exhibited during
the in-sample period as busts. Equity price busts which anticipate recessions are
much stronger compared to ordinary prices declines and end with the recession,
earliest though two quarters afterwards (see Claessens, Kose and Terrones (2009)
for detailed information).
The artificial indices are in accordance to equations (1)-(2) constructed with
0.0040,0.0080,...,0.2000δ = (i.e. corresponding to 1%,2%,...,50%annualδ = in
annual terms) uniformly distributed over time so that 50 different asset allocations
could be estimated which is also in line with Alexander and Dimitriu (2005).
Exhibit 1 gives an overview concerning the statistical properties, whereas in the
appendix, the asset allocations are given with respect to all estimated portfolios.
The optimization procedure concerning equation (7) suggest an asset allocation
corresponding to portfolio 29 (see tables 1a-d and figure 2) which tracks an
artificial index being constructed with 0.1160δ = (i.e. 29% in annual terms).
The three main positions join 94.40% of the overall weight allocation and are
invested in the industrial machinery sector, heavy electrical equipment industry
5 See IfW Press Release September 11, 2008.
K. Grobys 67
and the clothing industry. However, optimizing with respect to equation (8)
suggests a more diversified asset allocation. Portfolio 14 tracking an artificial
index with 0.0560δ = (i.e. 14% in annual terms, see tables 1a-d) invests only
40.53% in the same business sectors as portfolio 29.
The OMX 30 declined from September 12, 2008 until March 12, 2009 by
17.08%, whereas portfolio 14 declined only by 9.52% and performed
consequently 7.56% better in comparison to the benchmark while exhibiting a
marginal lower volatility of 53.50% p.a. compared to the benchmark’s volatility
being 53.51%. Exhibit 1 shows that the beta is close to the market beta
(i.e. 14ˆ 0.98β = ). Portfolio 29 which is optimal with respect to the optimization
procedure concerning equation (7) exhibits within the out-of-sample window a
loss of only 3.04% (i.e. corresponding to excess returns of 28.08% p.a.) while the
volatility is 7.99 per cent units higher in comparison to the benchmark’s volatility.
Again, the OLS regression being in line with the ordinary index model indicates
that the beta (i.e. 29ˆ 1.03β = ) is quite close to the market beta.
Testing for cointegration shows that all estimated weight allocations exhibit a
cointegration relationship with the artificial indices up to an enhancement factor
equal to 0.1320δ = (see table 2 in the appendix) on a 5% significance level.
Against it, portfolios tracking artificial indices involving trends being larger
35%annualδ = do not exhibit a cointegration relationship with the artificial
benchmark. Exhibit 1 shows that the abnormal returns being estimated in
accordance to the ordinary index model (see equation (11)) are statistically not
significant concerning the out-of-sample period. As the integrated artificial indices
are via construction cointegrated with the benchmark (see equations (1) and ((2)),
it can be concluded that the estimated portfolio exhibit a cointegration relationship
with the ordinary benchmark, too, while involving a stationary trend switching at maxt the direction. Figure 2 shows clearly the statistical arbitrage opportunity
since on 77% of all days, the cointegration optimal portfolio 29 outperforms the
68 Market timing and statistical arbitrage
index while exhibiting an excess return equal to 14.04% after 124 trading days
out-of-sample. Furthermore portfolios 23 and 25 exhibit maximum statistical
arbitrage opportunities as their returns are 19% above the index returns or higher
with an empirical probability of 50.14%. Even if portfolio 29 which tracks an
artificial index being enhanced by the factor 0.1160δ = (i.e. 29% in annual
terms) is optimal with respect to its Reward-to-Risk ratio, it generates returns of
29% p.a. above the index or higher with an empirical probability of 45.53%. The
statistical arbitrage opportunities are in accordance to the definition of
Bondarenko (2003) with respect to portfolio 29 limited up to excess returns of 9%
as the empirical probability that the portfolio generates returns of 9% or higher
than the index is 50% for the latter figure.
Figure 3 plots the Reward-to-Risk ratios and shows an increasing trend on a
decreasing rate while after the maximum, corresponding to portfolio 29, the ratio
is declining. Estimating the forecast adequacy concerning the maximum likelihood
optimal weight allocations gives the results of equation (12). Thereby, equation
(12) takes only the elements 1,…,33 into account, as first, a visual inspection of
the vector δ clearly shows a changed slope between 0.1320δ = and
0.1360δ = . The second indicator for a break is that cointegration optimality does
only hold for the sample 0.0040δ = until 0.1320δ = (see table 2 in the
appendix). Therefore, equation (13) takes also into account the break in the